We should note that, as we analyzed in Remark REF , under (A6), we have obtained the existence and uniqueness of the solution to the fully coupled FBSDEP (REF ), which can guarantee the \(\beta \) -integrability in the following. From the backward equation in (REF ), \(y\)  is given by the right-continuous version of \(y_t=E\big [\phi (x_T)+\int _t^Tg(t,x,y,z,\tilde{z})ds|\mathcal {F}_t\big ]\) . Since \(\beta \ge 2\) , by the martingale representation theorem in [1]} for local square integrable martingale, \((z,\tilde{z})\)  corresponds to the unique pair of \(\mathcal {E}\) -predictable processes satisfying
\(E\Big [\phi (x_T)+\int _0^Tg(t,x,y,z,\tilde{z})ds\big |\mathcal {F}_t\Big ]=y_0+\int _0^tz_sdW_s+\int _0^t\int _{\mathcal {E}}\tilde{z}_{(s,e)}\tilde{N}(de,ds),\ P\mbox{-}a.s.,\) 
